Superb!

In The Einstein Theory of Relativity, Lillian Lieber presents a young person’s account of the mathematics of special and general relativity. To begin, she uses a metaphor that now has become standard ... a canoe crossing a river. The Lorentz contraction is developed directly from the algebra used to explain the canoe’s position-shift. Moving on to general relativity, Lieber incorporates Lorentz’ transformation into a broader mathematical purview ... that of geodesics. As with the special case, the new structure is shown to predict that length, mass, and time undergo contraction near light-speed. Proceeding further, the rudiments of tensor analysis are outlined: contravariant, covariant and mixed properties are defined, then the Christoffel symbol, and finally the “small g” metric on the space-time continuum which results in, β, the stress energy tensor. From Einstein’s principle of equivalence (i.e. gravity is equated with acceleration) Lieber shows how Christoffel symbol symmetry reduces Einstein’s generalized geodesic (i.e. capital G) to merely twenty small g’s. The power to handle previously unexplained phenomena is displayed by calculation of Mercury’s planetary orbit. In conclusion, she presents the prediction that light “bends” around stars, as confirmed by the Eddington expedition of 1926.

Guy LeFrancois in his Psychology For Teaching notes, “Srebnik and Elias (1993) suggest that one fruitful approach toward reducing school dropout rates might be to not only try to make school more attractive and meaningful to students, but to do so before problems become apparent” (47-48). For precocious teens, one way to pique interest might be to introduce them to The Einstein Theory of Relativity. Lieber invites everyone to follow along for sheer joy of learning as she develops the mathematics of relativity. The book sports deft delivery that constructs the necessary tools along the way by building on principles of high school algebra. However, the reader is never asked to guess what comes next or to answer awkward end-of-chapter questions, and her skillful exposition obviates memorization. Rather, sophisticated derivations are demystified and contextualized by her prose, which ferries the reader with effortless elegance across a great divide to arrive cogently at a conclusion that is breathtaking in scope.

Unlike so many, Lieber emphasizes ideas rather than math. She challenges the mind and yet guides the way. The reader is encouraged to explore their ability to abstract. Challenging and guiding ... so important to Lev Vygotsky ... are described by LeFrancois: “[T]he task of educators, parents, and others charged with the ‘upbringing/teaching’ of children is to arrange for … activities that lie within [the zone of proximal growth] ... activities that, by definition, are neither too difficult nor too simple and therefore lead to continued growth (106) … [T]he zone of proximal growth describes tasks that require support (that is, scaffolding)” (108). Lieber provides support even as she collaborates with the reader to undertake the activity of growing their world by achieving a new vista: their own notion of the meaning underlying the idea of general relativity. She scaffolds understanding … not merely the symbol pushing.

By contrast, academic textbooks generally betoken pedagogy with tacit assumptions: the paramount importance of problem-solving performance, the obligatory covering of “requisite” material, as well as more subtle aims such as to establish the text as an authority, and to defer to corporate “needs.” Sensitive teens see through hidden agendas that reek of ulterior motive and ego. However, they respond positively when presented with a text they perceive to be honest. (The Einstein Theory of Relativity went through at least fourteen printings.) David Tall, the researcher who edited The Psychology Of Advanced Mathematical Thinking and a forerunner of that field’s burgeoning literature, explains: “It is no longer viable, if indeed it ever was, to lay the failure of our students on their supposed stupidity, when now the reasons behind their difficulties may be seen to be in part due to the epistemological nature of mathematics and in part due to misconceptions by mathematicians of how students learn. … [W]e now know, somewhat furtively, that the acquiring of those skills may develop concept imagery that contains the seed of future conflict (251). … We cheated our students because we did not tell the truth about the way mathematics works, possibly because we sought the Holy Grail of mathematical precision, possibly because we rarely reflected on … [or] realized, the true ways in which mathematics operates” (255).

Typical “adult” scientific texts fail to strike a balance between mathematical symbols and the ideas conveyed, but LeFrancois remarks it is well known that genuine intellectual ability is to be found in ideas, rather than strictly in symbols. “[T]hose with high measured intelligence also tend to be the most advanced and the most accomplished verbally - and vice-versa” (62). … Piaget contends that … even the development of logical thought processes are dependent on verbal interaction” (102). Recent pragmatic studies cause Harel and Sowder to argue similarly: “[Y]ears of instruction that focus on the results in mathematics, rather than the reasons behind those results, can leave the impression that only the results are important …” (41). It is critical to assist students as they negotiate conceptual change, that is “… the kind of learning required when the new information to be learned comes in conflict with prior knowledge … acquired [based on] everyday experiences” (Vosniadou and Verschaffel, 445).

While informal, Lieber’s writing style is not prototypically narrative. It is more akin to the labeled style often found in engineering texts. The text of the book is thick sans serif font that is laid out in narrow columns for ease of reading. Occasional words in “all caps” punctuate the page for emphasis and the sentences divide according to phrasing. The overall effect is conversational rather than didactic, making it ideal for a curious teenager’s summer vacation reading. Each paragraph is captioned not only by subject matter, but also by intent. Labeled style has advantages of uniformity and clarity of structure (Bagchi and Wells, 13), and it makes Lieber’s writing much more accessible to the reader than would mere narration. Thus, her style is not in the mathematical register per se - the register in which definitions, theorems and proof steps are written (Bagchi and Wells, 3). Neither is it some variety of general scientific or academic register. Her tone is conversational rather than didactic, matter-of-fact rather than rhetorical, geared toward how educators discuss a subject’s generalities in the everyday rather than toward developing multiple perspectives in an abstract explanatory framework (advocated by Epp, 892).

From start to finish, Lieber’s presentation is designed to disaffect the reader of common sense notions that run interference against the reader’s analytical bent of mind. Leron and Hazzan report how crucial such a strategy is in their examination of the parallels between dual-process theory ( “lateralization” in LeFrancois, 45) and the heuristics-and-biases research of Kahneman and Tversky. “[O]ur cognition and behavior operate in parallel in two quite different modes … roughly corresponding to our common sense notions of intuitive and analytical thinking … The two systems differ mainly on the dimension of accessibility: how fast and how easily things come to mind.” (5). Misconception that leads to logical fallacy is more likely to be due to “distraction by irrelevant external clues of high accessibility.” The implication is that factors having to do with wording and context of scholastic materials and instruction can lead to apparently irrational cognition. Adolescents engaged in a state of ‘moratorium’ - consisting of conflict between the need to find an identity and the difficulties involved in doing so - are done a disservice by mathematics education that fails to acknowledge the ways in which it baffles students. Lillian Lieber foresaw that disenchanting students works against the advancement of science.

I first read this book in college, some 30 years ago. I am pleased to see it freely available. It remains absolutely the clearest and most concise introduction to special relativity and the tensor-based mathematics of general relativity I have ever seen. Most treatments of the subject for lay readers tend to the non-mathematical. This book gives you the math, and yet is easy to follow. It is a superb bit of writing.